Thank you so much for the replies! I am used to dealing with electronics that don't really care how much current you put through them - after looking into it more and taking a look at the diagram for the stepper motor, I can now see that, each pair of wires is just a coil - the more current you put through it, the hotter it will get. Without any regulation on the stepper driver side, I could have easily burnt out the motor - the driver ended up failing before the motor.
I do have a spare EasyDriver chip which I was going to use for another project! Given the max amperage of the motor (350 mA) and the ability for this chip to regulate amperage from 150mA/phase to 700mA/phase, I think this will work (I won't need the "Big Easy Driver" for this motor). It even looks like the pins on the EasyDriver line up nicely with the proto hat for a pi, making everything a nice tight package.
In terms of a fuse with the 7A battery - If I can regulate amperage on the board, would I need a fuse?
Thanks again for the help! I've learned more about stepper motors in the last 24 hours than I ever thought I would!